In the last few decades the study on algebraic varieties has been very active. Not only algebraic varieties itself but also the fertility of the structures over algebraic varieties are attracting the scholars of algebra and geometry. Moreover, as we can see in the research of the conformal field theory and the Calabi-Yau manifolds, the relationship with various fields including physics is getting closer. In this project, paying attention to the interrelation among actions of groups on algebraic varieties, Hodge theory and period maps of Kahler manifolds, various moduli spaces on algebraic varieties, conformal field theory on arithmetic varieties, K-theory and number theory, we tried to make great progress in studying algebraic variety, In addition to individual studies in the neighborhoods of investigators, we organized several conferences to design close communication between related fields and sent members of the project to relevant conferences.We could get the following excellent results : construction of the moduli space of parabolic stable sheaves, study and applications of its structure, construction of the moduli space of stable sheaves on prejective schemes that may be singular, conformal field theory from the mathematical viewpoint, constructions, deformations and mirror symmetries of Calabi-Yau manifolds, development and applications of Model-Weil lattices, study and applications of K3 surfaces, existence prpblem of the surfaces of general type, development of Mori theory.